Obtaining lasers with both high optical power and good beam quality simultaneously has always been a difficult task. Vertical External Cavity Surface Emitting Lasers (VECSELs) are modifications to Vertical Cavity Surface Emitting Lasers (VCSELs), where a semiconductor gain medium is sandwiched between distributed Bragg reflectors (DBRs). In these laser systems, laser emissions are observed perpendicular to the surface, giving them a relatively low beam divergence and symmetric beam profile. VCSELs are quite convenient devices for certain applications; however, high power and high beam quality cannot be achieved together with conventional edge- or surface-emitting semiconductor lasers. Thus, optically pumped VECSELs have received considerable attention recently. They provide excellent beam quality and relatively high output powers. They achieve these properties due to their one-sided external cavity mirrors, which elongate the cavity. The longer the cavity is, the bigger the modes that come out; thus, a reduced number of modes contribute to the high-power beam purity. In addition, a longer cavity causes a finer laser bandwidth. On the other hand, the external cavity in VECSELs causes extra bulk by reducing robustness, alignment, and size, weight, and power (SWAP). To reduce the bulk to increase operability, diffractive optical elements (DOEs) can be utilized in VECSELs.
New technologies are now sought as well as innovative approaches to use DOEs in VCSEL chips for high-power applications to compensate for very short cavity length. Until now, research has focused on the adaptability of DOEs to certain laser systems in the academic domain. However, none of this work resulted in serious investigation of the feasibility of fabricating a VECSEL with an integrated DOE, nor was the full potential of diffractive optics for an extra-improved quality high-power laser system researched by investigating the latest innovations in laser and computer-generated DOE literature, specifically, computer-generated holograms (CGHs).
A CGH-based Improved VECSEL Cavity (CIVC), shown conceptually in FIG. 1 is proposed. FIG. 1 shows a set of DOEs placed with buffer layers in a cascaded state. This concept of cascaded CGHs is introduced to obtain certain laser functions. The most important property of the CIVC design is the freedom of performance merits. This design of a wavefront modulating scheme allows several diffractive structures to function to give the desired performance merits, as desired. Principally, the CIVC can be used to shape the laser modes as though there is an external mirror; at the same time, it may work to shape the output (as a flattop, for example).
Today lasers are used in a wide range of diverse applications, such as optical fiber communication, optical digital recording, materials processing, biophotonics, spectroscopy, imaging, entertainment, and defense. Therefore, it is very important to model fundamental laser operation computationally. FIG. 2 shows a simple abstraction of a laser cavity resonator. Physically, electromagnetic radiation is reflected back and forth for the amplification of the radiation in a closed volume defined by a length D and an aperture a. In a broader sense, cavities have two main considerations: the stability criterion imposed by the end-reflector mirrors and the structure of modes created by the cavity geometry. The former basically sets limits on the sustainability of the amplification. Based on the ray model of geometrical optics under paraxial approximation, light rays experience periodic focusing. This effect can also be considered as an outcome caused by the continuum sequence of lenses. For the particular system shown, the stability condition coming from the ray model analysis simply yieldsR1>D>0  (1)
Equation (1) states that the cavity length must be smaller than the radius of curvature of the first mirror for a stable operation ensuring self-focusing. In fact, D is crucial for the operation of the device. Another important figure of merit is a, since it adjusts the volume of oscillations and is responsible from mode sizes and shapes. If the laser is optically pumped, the mirrors' coatings should be selected accordingly; also, the output mirror's (out-coupler: Mirror 2 in this case) coating should allow ˜10% transmittance for the resonant wavelength.
Basically, the Fox-Li algorithm states that in a Fabry-Perot resonator cavity, oscillating modes travel back and forth between mirrors and lose energy as they are being diffracted by apertures. After enough round-trips, the electric field becomes stable so that it repeats itself in each round-trip, yielding eigenvalues and Eigen functions of that specific cavity. By considering non-negative integers p and l that define mode numbers, an eigenvalue equation for a closed cavity can be written as{circumflex over (P)}Ψpl=γplΨpl  (2)
In Equation (2), {circumflex over (P)} is the round-trip propagation operator responsible from propagation of the field one round-trip. Eigenfunctions (Ψ) are the possible E-fields and can be solutions of the wave equation (subject to boundaries) with eigenvalues γpl. As a consequence of the principle of superposition, the total field U in a laser cavity can be expanded as linear combinations of these eigenfunctions.
                    U        =                              ∑                          p              ,              l                                ⁢                                    c              pl                        ⁢                                          Ψ                pl                            .                                                          (        3        )            
The constant cpl adjusts weights for eigenfunctions and depends mainly on the cavity shape as well as initial conditions. At the point of saturation, oscillating fields do not change shape from kth to (k+1)nth round-trip iteration. Then in an orthogonal space, one can define specific modes as follows.Ψpl(k+1)=(γpl)kΨpl(k)  (4)
As the way these modes' relative strength (with respect to each other) is expressed by cpl, their evolution with respect to the round-trip operator is given by the constant γpl. Equation (4) implies a change between round-trips. The constant depends on the apertures and diffraction losses coming from these apertures. Although this equation predicts the disappearance of every mode eventually, the power feedback from the laser prevents that. Since lower order mode sizes are smaller, they are favored to propagate. As a consequence, γ00 is expected to be the largest with respect to the other mode constants. This corresponds to the fundamental TEM00 mode, with Gaussian shape. In fact, although lasers fundamentally tend to oscillate in higher index modes based on the supplied energy, due to diffraction and absorption effects at apertures, intensities of the higher modes may gradually be dissipated and disappear. The issue of how many modes are allowed depends on the aperture size; the larger the aperture, the higher the modes that can be excited, thus more power goes into these modes.
To determine the available set of modes in a resonator, the cavity is simulated by defining each element as an operator, considering the cavity as a continuum that provides successive propagation, and exchanging mirrors with lenses doing the same function. As a consequence, This system reaches equilibrium, demonstrating the output of the laser with its constituent mode patterns. Operator notation is given in Table 1 for some common cavity elements.
In Table 1, {circumflex over (L)}, Â, and Ĥ operators are shown. Note that the propagation operator is chosen based on the relation between propagation distance (equal to the cavity length, D), sampling interval, ρ, and feature size length of the medium under consideration, L. The spatial frequencies (fx,fy) are computed on a grid from max to min with steps 1/L. From the Nyquist-Shannon sampling theorem, fmax and fmin, are ±1/(2ρ).
The final operator {circumflex over (P)} is found by multiplying these individual operators by following the sequence. For example, a simple resonator starts with a lens covered by an aperture, then propagation in the cavity, followed by a planar mirror that is just represented by a simple aperture again, and finally propagation back to the starting point. Thus, we have {circumflex over (P)}=ĤÂĤÂ{circumflex over (L)}. The operator order goes from left to right. The operator Ĥ is applied in the Fourier domain; this means that the field right before applying the propagator is transformed, multiplied with the relevant Ĥ, then inverse transformed back to the spatial domain. All operators and fields can be thought of as matrices; all multiplications are element-wise matrix multiplications. Therefore, successive application of {circumflex over (P)} yields the desired output mode pattern. Some simulation results can be seen in FIG. 3(a)-(d).
TABLE 1Common OperatorsOperatorExplanation      L    ^    =      exp    ⁡          [                                    -            ik                                R            1                          ⁢                  (                                    x              2                        +                          y              2                                )                    ]      Lens function, creates focusing effect as a concave mirror would do. k is the wave-number.       A    ^    =            rect      ⁡              (                  x          a                )              ⁢                  ⁢          rect      ⁡              (                  y          a                )            Aperture function for rectangular geometry.       A    ^    =      circ    ⁡          (                        r          2                          a          /          2                    )      Aperture function for cylindrical geometry. H = exp[−iπλD(fx2 + fy2)]Free-space propagator for a distance z, if z ≤ ρL/λ.       H    ^    =          ⁢        ⁢                  ⁢          {                        1                      i            ⁢                                                  ⁢            λD                          ⁢                  exp          ⁡                      [                                                            -                  i                                ⁢                                                                  ⁢                π                                            λD                ⁡                                  (                                                            x                      2                                        +                                          y                      2                                                        )                                                      ]                              }      Free-space propagator for a distance z, if z > ρL/λ.   denotes Fourier transform.
FIG. 3(a) shows a rectangular cavity aperture example. In simulations, L indicates the total area assigned, which always contains 64×64 pixels. The aperture is a, and always contains 40×40 pixels. While its value is increased step-by-step, the increment in mode numbers increases as a general trend. D is always 15 cm. FIG. 3(b)-(d) show three results: on the left-hand side, a TEM00 or Gaussian beam for a=1 mm, with diameter 0.8 mm; in the middle, a TEM22; and on the right-hand, side a collection of modes. Gaussian beams were examined because of their well-behaved long-distance propagation. The intensity pattern is defined in Equation (5).
                                          I            ⁡                          (                              r                ,                z                            )                                =                                                                      I                  0                                ⁡                                  (                                                            w                      0                                                              w                      ⁡                                              (                        z                        )                                                                              )                                            2                        ⁢                          exp              ⁡                              [                                  -                                                            2                      ⁢                                              (                                                                              x                            2                                                    +                                                      y                            2                                                                          )                                                                                                            w                        ⁡                                                  (                          z                          )                                                                    2                                                                      ]                                                    ,                            (        5        )                        where                                                                w          ⁡                      (            z            )                          =                              w            0                    ⁢                                                    1                +                                                      (                                          z                                              z                        R                                                              )                                    2                                                      .                                              (        6        )            
The constant zR is Rayleigh length, and is equal to πw02/λ. Therefore, in free-space propagation of the Gaussian beam, the beam waist is minimum when z is zero and is equal to w0 according to Equation (6). Minimum beam waist is a constant, and the location depends on the laser. If there is a lens, w0 is observed at the focal point. As the beam propagates, beam waist w is actually the radius at which the intensity drops to 1/e2 times that of the central maximum, and it increases according to Equation (6). Throughout this study, we measured beam diameters with 2w by calling them the beam waist (although more correctly this is a diameter) between the 1/e2 power points Rectangular geometry is principally displayed, however cylindrical geometry gives similar trends.